If M is a `3xx3` matrix such that `M^(2)=O`, then det. `((I+M)^(50)-50M)` where I is an identity matrix of order 3, is equal to:

Options

A. 3
B. 50
C. 2
D. 1

Explanation

To solve the problem, we need to find the determinant of the expression \((I + M)^{50} - 50M\) given that \(M\) is a \(3 \times 3\) matrix such that \(M^2 = O\) (the zero matrix).

### Step-by-step Solution:

1. **Understanding the Matrix \(M\)**:
Since \(M^2 = O\), this means that \(M\) is a nilpotent matrix. For a \(3 \times 3\) matrix, this implies that \(M\) can have at most 2 non-zero eigenvalues, and all eigenvalues of \(M\) are 0.

2. **Calculating \((I + M)^2\)**:
\[ (I + M)^2 = (I + M)(I + M) = I^2 + IM + MI + M^2 = I + M + M + O = I + 2M \]

3. **Calculating \((I + M)^3\)**:
\[ (I + M)^3 = (I + M)(I + 2M) = I(I + 2M) + M(I + 2M) = I + 2M + M + 2M^2 = I + 3M + O = I + 3M \]

4. **Generalizing \((I + M)^n\)**:
From the pattern observed, we can generalize:
\[ (I + M)^n = I + nM \]
This holds true for \(n = 1, 2, 3\) and can be proven by induction.

5. **Calculating \((I + M)^{50}\)**:
Using the generalized formula:
\[ (I + M)^{50} = I + 50M \]

6. **Finding the Expression**:
We need to compute:
\[ (I + M)^{50} - 50M = (I + 50M) - 50M = I \]

7. **Calculating the Determinant**:
The determinant of the identity matrix \(I\) is:
\[ \det(I) = 1 \]

### Final Answer:
Thus, the value of \(\det((I + M)^{50} - 50M)\) is:
\[ \boxed{1} \]